+1725 Let p(x) be a cubic polynomial. If p(0) = 0, p(1) = 1, p(2) = 2, and p(3) = 3, then compute p(4).
+130466 Let p(x) = ax^3 + bx^2 + cx + d
If p(0) = 0, then d = 0
So we have the equations
a(1)^3 + b(1)^2 + c(1) = 1
a(2)^3 + b(2)^2 + c(2) = 2
a(3)^3 + b(3)^2 + c(3) =3 simplify these
a + b + c = 1 (1)
8a +4b + 2c = 2 (2)
27a + 9b + 3c = 3 (3)
Multiply (1) by -2 add to (2) and multiply 1 by -3 and add to (3)
6a + 2b = 0 → 3a + b = 0 → -3a - b = 0 ( 4)
24a + 6b = 0 → 4a + b = 0 (5)
Add (4) , (5) a = 0 and so b = 0
No such cubic polynomial exists
